Problem 86
Question
Prove the identity. $$\frac{\sin 3 x+\sin 7 x}{\cos 3 x-\cos 7 x}=\cot 2 x$$
Step-by-Step Solution
Verified Answer
The identity \(\frac{\sin 3x + \sin 7x}{\cos 3x - \cos 7x} = \cot 2x\) is verified using trigonometric identities.
1Step 1: Apply Sum-to-Product Formulas
The sum-to-product identities allow us to express sums and differences of sines or cosines as products. The identity for the sum of sines is: \[\sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right)\]Applying this to the numerator \(\sin 3x + \sin 7x,\)we get\[\sin 3x + \sin 7x = 2 \sin\left(\frac{3x+7x}{2}\right) \cos\left(\frac{3x-7x}{2}\right) = 2 \sin 5x \cos(-2x) = 2 \sin 5x \cos 2x.\]
2Step 2: Apply Difference-to-Product Formulas
The difference of cosines identity is:\[\cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right)\]Applying this to the denominator \(\cos 3x - \cos 7x,\)we get\[\cos 3x - \cos 7x = -2 \sin\left(\frac{3x+7x}{2}\right) \sin\left(\frac{3x-7x}{2}\right) = -2 \sin 5x \sin(-2x) = 2 \sin 5x \sin 2x.\]
3Step 3: Simplify the Expression
Substitute the transformed numerator and denominator into the original expression:\[\frac{2 \sin 5x \cos 2x}{2 \sin 5x \sin 2x}\]Cancel the common factor \(2 \sin 5x\) in the numerator and denominator:\[\frac{\cos 2x}{\sin 2x}\]
4Step 4: Recognize the Cotangent Function
The expression \(\frac{\cos 2x}{\sin 2x}\) is the definition of the cotangent function. Thus, the expression simplifies to:\[\cot 2x\]This completes the proof of the identity.
Key Concepts
Sum-to-Product FormulaDifference-to-Product FormulaCotangent Function
Sum-to-Product Formula
Understanding trigonometric identities like the sum-to-product formula can make simplifying expressions much easier. The sum-to-product identities are particularly useful when you have an addition of sine or cosine terms. These identities transform a sum of trigonometric functions into a product, which often simplifies the calculations. For instance, if you're dealing with
Let's see this in action: Say we have the expression \( \sin 3x + \sin 7x \). Using the formula, you would get
This technique is helpful because products are often easier to work with or further simplify than sums, particularly in solving trigonometric equations or proving identities.
- \( \sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \)
Let's see this in action: Say we have the expression \( \sin 3x + \sin 7x \). Using the formula, you would get
- \( \sin 3x + \sin 7x = 2 \sin 5x \cos 2x \)
This technique is helpful because products are often easier to work with or further simplify than sums, particularly in solving trigonometric equations or proving identities.
Difference-to-Product Formula
The difference-to-product formulas work similarly to the sum-to-product formulas but are applied when you have subtraction instead of addition. These identities help convert differences of sines or cosines into products, which simplify expressions and make further manipulation easier. Consider the formula:
Utilizing this identity with the expression \( \cos 3x - \cos 7x \), you achieve:
Using difference-to-product conversions is valuable because, in many problems, finding a product-form simplifies the division process, especially if it involves canceling common factors.
- \( \cos A - \cos B = -2 \sin \left( \frac{A+B}{2} \right) \sin \left( \frac{A-B}{2} \right) \)
Utilizing this identity with the expression \( \cos 3x - \cos 7x \), you achieve:
- \( \cos 3x - \cos 7x = 2 \sin 5x \sin 2x \)
Using difference-to-product conversions is valuable because, in many problems, finding a product-form simplifies the division process, especially if it involves canceling common factors.
Cotangent Function
The cotangent function is a fundamental trigonometric function defined as the reciprocal of the tangent function. For any angle \( \theta \), the cotangent is given by
In the context of simplifying or proving trigonometric identities, recognizing the expression \( \frac{\cos \theta}{\sin \theta} \) as the cotangent is crucial. For example, if you transform an expression to \( \frac{\cos 2x}{\sin 2x} \), it immediately recognizes as \( \cot 2x \). This concept allows you to conclude or verify that an expression simplifies correctly, such as proving that the identity \( \frac{\sin 3x + \sin 7x}{\cos 3x - \cos 7x} = \cot 2x \) holds true.
Cotangent is especially useful in trigonometry when dealing with right triangles, circles, or periodic functions.
- \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \)
In the context of simplifying or proving trigonometric identities, recognizing the expression \( \frac{\cos \theta}{\sin \theta} \) as the cotangent is crucial. For example, if you transform an expression to \( \frac{\cos 2x}{\sin 2x} \), it immediately recognizes as \( \cot 2x \). This concept allows you to conclude or verify that an expression simplifies correctly, such as proving that the identity \( \frac{\sin 3x + \sin 7x}{\cos 3x - \cos 7x} = \cot 2x \) holds true.
Cotangent is especially useful in trigonometry when dealing with right triangles, circles, or periodic functions.
Other exercises in this chapter
Problem 85
Prove the identity. $$\frac{\sin x+\sin 5 x}{\cos x+\cos 5 x}=\tan 3 x$$
View solution Problem 85
Verify the identity. $$\frac{\sin ^{3} x+\cos ^{3} x}{\sin x+\cos x}=1-\sin x \cos x$$
View solution Problem 86
Verify the identity. $$\frac{\tan v-\cot v}{\tan ^{2} v-\cot ^{2} v}=\sin v \cos v$$
View solution Problem 87
Prove the identity. $$\frac{\sin 10 x}{\sin 9 x+\sin x}=\frac{\cos 5 x}{\cos 4 x}$$
View solution